Multi-scaling of moments in stochastic volatility models

TL;DR

This paper introduces stochastic volatility models exhibiting multi-scaling of moments, where the scaling behavior of increments changes at a threshold, capturing complex phenomena observed in financial time series.

Contribution

It characterizes conditions under which multi-scaling occurs, linking it to superlinear mean reversion driven by Levy subordinators with power law tails.

Findings

Multi-scaling observed in financial asset data.

Multi-scaling linked to superlinear mean reversion.

Conditions established for multi-scaling based on Levy process properties.

Abstract

We introduce a class of stochastic volatility models $(X_t)_{t \geq 0}$ for which the absolute moments of the increments exhibit anomalous scaling: $\E\left(|X_{t+h} - X_t|^q \right)$ scales as $h^{q/2}$ for $q < q^*$, but as $h^{A(q)}$ with $A(q) < q/2$ for $q > q^*$, for some threshold $q^*$. This multi-scaling phenomenon is observed in time series of financial assets. If the dynamics of the volatility is given by a mean-reverting equation driven by a Levy subordinator and the characteristic measure of the Levy process has power law tails, then multi-scaling occurs if and only if the mean reversion is superlinear.